To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein-spin-fermion model. In SLQMC, machine learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic state to the charge-density-wave state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared-error of the machine learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared-error as the machine learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.
To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein-spin-fermion model. In SLQMC, machine learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic state to the charge-density-wave state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared-error of the machine learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared-error as the machine learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.
To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein spin-fermion model. In SLQMC, machine-learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic (AFM) state to the charge-density-wave (CDW) state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared error of the machine-learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared error as the machine-learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine-learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.
The self-learning Monte Carlo method (SLMC), using a trained effective model to guide Monte Carlo sampling processes, is a powerful general-purpose numerical method recently introduced to speed up simulations in (quantum) many-body systems. In this Rapid Communication, we further improve the efficiency of SLMC by enforcing physical symmetries on the effective model. We demonstrate its effectiveness in the Holstein Hamiltonian, one of the most fundamental many-body descriptions of electron-phonon coupling. Simulations of the Holstein model are notoriously difficult due to a combination of the typical cubic scaling of fermionic Monte Carlo and the presence of extremely long autocorrelation times. Our method addresses both bottlenecks. This enables simulations on large lattices in the most difficult parameter regions, and an evaluation of the critical point for the charge density wave transition at half filling with high precision. We argue that our work opens a research area of quantum Monte Carlo, providing a general procedure to deal with ergodicity in situations involving Hamiltonians with multiple, distinct low-energy states.
The Self-Learning Monte Carlo (SLMC) method is a Monte Carlo approach that has emerged in recent years by integrating concepts from machine learning with conventional Monte Carlo techniques. Designed to accelerate the numerical study of interacting many-body systems, SLMC significantly improves sampling efficiency by constructing an effective model -- via machine learning methods -- based on configurations generated by conventional Monte Carlo methods and then proposes global updates based on the effective model. This enhancement leads to a substantial reduction in autocorrelation time, especially near the critical region, where traditional methods typically suffer from critical slowing down and increased computation complexity. Moreover, SLMC maintains statistical accuracy by implementing a cumulative update scheme that rigorously satisfies the detailed balance condition. And more recent applications have extended the SLMC to convolutional neural networks with applications not only in condensed matter physics but also high-energy physics, quantum chemistry, and quantum simulations. The generic applicability and high computational efficiency make SLMC a powerful and scalable framework for quantum Monte Carlo simulations of strongly correlated electron systems, extending the reach of numerical investigations beyond the limitations of conventional techniques.
Monte Carlo simulations are crucial for examining the Ising model, especially when it's tough to find analytical solutions. However, traditional Monte Carlo methods, like the Metropolis algorithm, encounter significant hurdles, such as slowing down near phase transitions and issues related to finite sizes. This paper looks into both the advantages and limitations of these traditional Monte Carlo techniques. It also covers recent developments like Tensor Network Monte Carlo and Quantum Monte Carlo methods, which have shown promise in overcoming these challenges. Furthermore, the paper explores how machine learning techniques are being incorporated into Monte Carlo simulations to enhance their efficiency and accuracy. These advancements represent a major leap forward in simulating complex systems and have expanded the use of Monte Carlo methods into new areas, including quantum systems and models with disorder. This paper finds that traditional Monte Carlo methods, like the Metropolis algorithm, face issues such as critical slowing down and finite-size effects. New methods, such as Tensor Network Monte Carlo (TNMC) and Quantum Monte Carlo (QMC), effectively address these challenges, enhancing simulation accuracy and efficiency in complex systems. Additionally, integrating machine learning optimizes parameters and accelerates convergence, expanding Monte Carlo’s applicability to large-scale, high-dimensional systems across various scientific fields.
We employ the ``learning by confusion'' technique, an unsupervised machine learning approach for detecting phase transitions, to analyze quantum Monte Carlo simulations of the two-dimensional Holstein model---a fundamental model for electron-phonon interactions on a lattice. Utilizing a convolutional neural network, we conduct a series of binary classification tasks to identify Holstein critical points based on the neural network's learning accuracy. We further evaluate the effectiveness of various training datasets, including snapshots of phonon fields and other measurements resolved in imaginary time, for predicting distinct phase transitions and crossovers. Our results culminate in the construction of the finite-temperature phase diagram of the Holstein model.
Abstract We propose the use of the ‘spin-opstring’, derived from Stochastic Series Expansion quantum Monte Carlo (QMC) simulations as machine learning (ML) input data. It offers a compact, memory-efficient representation of QMC simulation cells, combining the initial state with an operator string that encodes the state’s evolution through imaginary time. Using supervised ML, we demonstrate the input’s effectiveness in capturing both conventional and topological phase transitions, and in a regression task to predict non-local observables. We also demonstrate the capability of spin-opstring data in transfer learning by training models on one quantum system and successfully predicting on another, as well as showing that models trained on smaller system sizes generalize well to larger ones. Importantly, we illustrate a clear advantage of spin-opstring over conventional spin configurations in the accurate prediction of a quantum phase transition. Finally, we show how the inherent structure of spin-opstring provides an elegant framework for the interpretability of ML predictions. Using two state-of-the-art interpretability techniques, Layer-wise Relevance Propagation and SHapley Additive exPlanations, we show that the ML models learn and rely on physically meaningful features from the input data. Together, these findings establish the spin-opstring as a broadly-applicable and interpretable input format for ML in quantum many-body physics.